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Edited Linear Algebra 101

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Mark 2023-04-10 11:48:29 -07:00
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Advanced/Linear Algebra 101/main.tex
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@ -107,37 +107,42 @@
\end{tikzpicture}
\end{center}
\vfill
\problem{}
If $a$ and $b$ are perpendicular, what must $\langle a, b \rangle$ be? Is the converse true?
\vfill
\pagebreak
\begin{minipage}[t]{0.48\textwidth}\vspace{0pt}
\begin{center}
\begin{tikzpicture}[scale=1]
\draw[dashed,->] (-0.5,0) -- (4,0) node[right]{};
\draw[dashed,->] (0,-0.5) -- (0,3) node[above]{};
\section{Bonus}
\draw[->] (0,0) -- (1,2) node[right]{};
\draw[->] (0,0) -- (3,0.5) node[above]{};
\end{tikzpicture}
\end{center}
\end{minipage}
\begin{minipage}[t]{0.48\textwidth}\vspace{0pt}
\begin{center}
\begin{tikzpicture}[scale=1]
\draw[dashed,->] (-0.5,0) -- (4,0) node[right]{};
\draw[dashed,->] (0,-0.5) -- (0,3) node[above]{};
\problem{}
Show that the euclidean norm satisfies the triangle inequalty:
$$
||x+y|| \leq ||x|| + ||y||
$$:
\draw[->] (0,0) -- (3,1) node[right]{};
\draw[->] (0,0) -- (3,0.5) node[above]{};
\end{tikzpicture}
\end{center}
\end{minipage}
\vfill
\problem{}
Show that the eucidean norm satisfies the reverse triangle inequality:
$$
||x - y|| \geq |~||x|| - ||y||~|
$$
\vfill
\problem{}
Prove the Cauchy-Schwartz inequality:
$$
||\langle x, y \rangle|| = ||x||~||y||
$$
\vfill
\end{document}